Homomorphisms from Specht Modules to Signed Young Permutation Modules

Abstract

We construct a class ΘR of homomorphisms from a Specht module SλZ to a signed permutation module MZ(α|β), which generalises James's construction of homomorphisms whose codomain is a Young permutation module. We show that any ϕ ∈ HomZSn(SλZ, MZ(α|β)) lies in the Q-span of Θsstd, a subset of ΘR corresponding to semistandard λ-tableaux of type (α|β). We also study the conditions for which ΘFsstd-a subset of HomFSn(SλF, MF(α|β)) induced by Θsstd-is linearly independent, and show that it is a basis for HomFSn(SλF, MF(α|β)) when FSn is semisimple.